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Full text of "Treatise On Analysis Vol-Ii"

298        XIV    INTEGRATION IN LOCALLY COMPACT GROUPS

3. Let G be a locally compact group, /? a left Haar measure on G. If G = {*?}, the
algebra L*(G, j8) has zero-divisors =0. To construct two nonnegligible functions
/, g in ^'(G, /?) such that/*# is negligible, we may proceed as follows:

(1)    The case where G has a compact subgroup H ^ {e}. Take for/a character-
istic function q?A and for g a function of the form <psQ  9?e, where A, B, and s are
suitably chosen, and remark that AG(*) = 1 for all x e H.

(2)    The case G = Z. Show that we may take

'

2/1-1     2/7+1

for all n E Z, and g = /.

(3) The general case. Prove first of all that there exists a ^ e in G such that
A(fl) = 1. The closure H in G of the subgroup generated by a is then either compact or
isomorphic to Z (Section 12.9, Problem 10). In the former case, use the result of case
(1); in the latter, take

+ 00                                                                           oo

/(/)==     2^    an<pua-n(0>       #(0 =     ^L    Pn'jPani/O

n= - oo                                                           n- - oo

where the set U and the sequences (), (ft) are suitably chosen with the help of
case (2).

4. Let G be a locally compact group, f$ a left Haar measure on G. In order that a subset
H of -S?P(G, ft) (1 ^p < H- oo) should have a relatively compact image ft in LP(G, J8),
it is necessary and sufficient that the following conditions should be satisfied: (1) ft
is bounded in LP(G, /?); (2) for each  > 0, there exists a compact subset K of G such
that Np(/9G-ic) ^ e for all/e H; (3) for each e > 0, there exists a neighborhood V
of e in G such that NP((Y(.S)/) f)^e for all/e H and all s e V. (To prove that these
conditions are sufficient, observe that if g G JT(G) and if L is a compact subset of G,
then the image, under the mapping/i*g */, of the set of restrictions to L of functions
belonging to H is an equicontinuous subset of .

5. Let G be a locally compact group, j8 a left Haar measure on G, and ft a positive
measure on G. Show that if A is a //,-integrable set and B is a universally measurable
set in G, the function u : s\>/x(A n sB) is ^-measurable on G. If moreover B~l is
/3-integrable, then so is u and we have

(Use the Lebesgue-Fubini theorem and Problem 21 of Section 13.9.) Give an example
of a measure ft such that the function s-i /x(A n .sB) is not continuous. If ft is a
measure with basis /3 and if A is /u-integrable and B is universally measurable, then
the function s\ >/x(A n 56) is continuous on G.

6. Let G be a locally compact group, G' a topological group, /a homomorphism of G
into G' which is ^-measurable, where ft is a left Haar measure on G, Show that /is
continuous. (Observe that if we put #(#) =/(jc~l), there exists a compact non--
negligible subset K of G such that the restrictions of /and g to K are continuous.
Deduce that the restriction of /to K  K"1 is continuous, by using (12.3.8); then apply
(14.10.8).) >7/?/'r,\res (/An * (/  j8)) converges to (/u,B * (/  j8)) with respect to ^6